# Geometric Brownian Motion with Nonuniform Time Grid

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This Demonstration simulates geometric Brownian motion (GBM) paths with a nonuniform time grid. A GBM is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. In computational finance, GBM is used to model stock prices in the Black–Scholes model and is the most widely used model of stock price behavior.

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Contributed by: Michail Bozoudis (June 2016)

Suggested by: Michail Boutsikas

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Let denote the number of time steps and the number of GBM paths. The list of the stock prices that the GBM path consists of are generated according to the following process:

, , where , follows the standard normal distribution, and is the drift of the stochastic process.

The Monte Carlo estimation of the European put option derives from the discounted average of the option's possible payoffs at expiry:

.

This Demonstration does not use Mathematica's built-in function GeometricBrownianMotionProcess because the Monte Carlo simulation using RandomFunction requires fixed time steps as in RandomFunction[GeometricBrownianMotionProcess[mu,sigma,x0],{t0,tend,dt}].

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